Power Concavity of Low-Dimensional Unimodular Fiber Counts
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Abstract
We prove weighted power-concavity inequalities for lattice-point counts of bounded unimodular fibers. In central affine dimension at most two, the square root of the corrected count N − 1/4 satisfies weighted Jensen; the correction 1/4 is best possible. If every endpoint count is at least m ≥ 1, the power (2m − 1)/(4m − 1) is valid. The best universal power obeys A(m) = 1/2 − 1/(8m) + O(m^(−3/2)), with explicit finite-threshold bounds. The obstruction uses a fixed six-edge network. We also prove that 4/13 is the optimal concavity power of the classical continuous piecewise polynomial extension of the type-A₃ Kostant partition function. The planar argument combines unimodular elimination, Pick’s theorem and classical Brunn–Minkowski concavity; the A₃ argument uses the classical chamber formulas and a Hessian comparison. These results give restricted global Jensen inequalities.
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Research details
- Subjects
- Mathematics · Combinatorics and Graph Theory
- Keywords
- unimodular fibers, integer flows, power concavity, lattice-point counts, Kostant partition function, Brunn–Minkowski inequality
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
The corrected square-root and count-dependent power inequalities assume bounded nonempty unimodular fibers with central affine dimension at most two and admissible finite weighted integral demands. The leading asymptotic deficit is sharp; exact optimal powers at finite thresholds remain undetermined. The 4/13 optimality result applies to the specified classical continuous type-A₃ extension.
Disclosures
AI-assisted tools were used in developing and internally checking the arguments and preparing the manuscript. Classical convex geometry and chamber enumeration are explicitly credited.
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References and related resources
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